In probability a quasi-stationary distribution is a random process that admits one or several absorbing states that are reached almost surely, but is initially distributed such that it can evolve for a long time without reaching it. The most common example is the evolution of a population: the only equilibrium is when there is no one left, but if we model the number of people it is likely to remain stable for a long period of time before it eventually collapses.
Formal definition We consider a Markov process ( Y t ) t ≥ 0 {\displaystyle (Y_{t})_{t\geq 0}} taking values in X {\displaystyle {\mathcal {X}}} . There is a measurable set X t r {\displaystyle {\mathcal {X}}^{\mathrm {tr} }} of absorbing states and X a = X ∖ X tr {\displaystyle {\mathcal {X}}^{a}={\mathcal {X}}\setminus {\mathcal {X}}^{\operatorname {tr} }} . We denote by T {\displaystyle T} the hitting time of X tr {\displaystyle {\mathcal {X}}^{\operatorname {tr} }} , also called killing time. We denote by { P x ∣ x ∈ X } {\displaystyle \{\operatorname {P} _{x}\mid x\in {\mathcal {X}}\}} the family of distributions where P x {\displaystyle \operatorname {P} _{x}} has original condition Y 0 = x ∈ X {\displaystyle Y_{0}=x\in {\mathcal {X}}} . We assume that X tr {\displaystyle {\mathcal {X}}^{\operatorname {tr} }} is almost surely reached, i.e. ∀ x ∈ X , P x ( T < ∞ ) = 1 {\displaystyle \forall x\in {\mathcal {X}},\operatorname {P} _{x}(T<\infty )=1} . The general definition is: a probability measure ν {\displaystyle \nu } on X a {\displaystyle {\mathcal {X}}^{a}} is said to be a quasi-stationary distribution (QSD) if for every measurable set B {\displaystyle B} contained in X a {\displaystyle {\mathcal {X}}^{a}} , ∀ t ≥ 0 , P ν ( Y t ∈ B ∣ T > t ) = ν ( B ) {\displaystyle \forall t\geq 0,\operatorname {P} _{\nu }(Y_{t}\in B\mid T>t)=\nu (B)} where P ν = ∫ X a P x d ν ( x ) {\displaystyle \operatorname {P} _{\nu }=\int _{{\mathcal {X}}^{a}}\operatorname {P} _{x}\,\mathrm {d} \nu (x)} . In particular ∀ B ∈ B ( X a ) , ∀ t ≥ 0 , P ν ( Y t ∈ B , T > t ) = ν ( B ) P ν ( T > t ) . {\displaystyle \forall B\in {\mathcal {B}}({\mathcal {X}}^{a}),\forall t\geq 0,\operatorname {P} _{\nu }(Y_{t}\in B,T>t)=\nu (B)\operatorname {P} _{\nu }(T>t).}
General results
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